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Stabilization by diffusion

\includegraphics[width = 0.75\textwidth]{doelman}

Figure 1: The homoclinic pulse solution of (4) for $ \varepsilon ^2 = 0.1$ and $ \mu = 0.38$. Here, the scaled pulses $ \varepsilon U_h(x)$ and $ \varepsilon V_h(x)$ are plotted on the long spatial scale $ X = \varepsilon x$. The singular `sharp' pulse represents the $ U$-component, it is $ {\cal O}(\varepsilon )$ close to a homoclinic solution of the scalar equation (1) with $ F(U) = \frac{2}{\sqrt{\mu}}U^2 - U$ (Theorem 1). This figure has been obtained by direct numerical simulation of (4) using the moving-grid code described in [2].

Although pulse solutions of scalar equations can thus never be stable, there are many systems of PDEs, such as the Gray-Scott model for autocatalytic reactions [7] and the Gierer-Meinhardt model, that exhibit pulse solutions that are very similar to scalar pulses, see Figure 1. These systems can be written in the following form

\begin{displaymath} \left\{ \begin{array}{rcccr} U_t & = & U_{xx} & + & G(U, V)... ...t & = & V_{xx} & + & \varepsilon ^2 H(U, V) \end{array}\right. \end{displaymath} (3)

where $ 0 < \varepsilon \ll 1$ is an asymptotically small parameter. Since both $ U(x,t)$ and $ V(x,t)$ are assumed to be bounded for $ x \in {\bf R}$ - a natural condition for solutions to PDEs on unbounded domains - it follows that $ V(x,t) \to V_0$, a constant, as $ \varepsilon \to 0$. Hence, (3) reduces to (1), with $ F(U) = G(U, V_0)$ in the limit $ \varepsilon \to 0$. Note that a coupled system of reaction-diffusion equations in which the ratio of the diffusion coefficients can be assumed to be small, as is the case for the Gray-Scott and the Gierer-Meinhardt model, can always be written in the form (3).

The limit $ \varepsilon \to 0$ intuitively yields two conclusions for the situation $ 0 \neq \varepsilon \ll 1$: (i) it can be expected that (3) indeed has solutions $ (U_h(x), V_h(x))$ of pulse type of which the $ U$-component $ U_h(x)$ is close to a pulse solution $ u_h(x)$ of a scalar PDE (1); (ii) the pulse solution $ (U_h(x), V_h(x))$ can also not be stable: the spectrum of the linearization around $ (U_h(x), V_h(x))$ should be close to that of the scalar case (which has an $ {\cal O}(1)$ unstable eigenvalue). In general, neither of these assertions is (completely) true. Moreover, both the existence (i) and the stability (ii) question require the development of novel ideas and techniques. For instance, the scalar problem (1) with $ F(U) = G(U, V_0)$ will have a pulse solution for open $ V_0$-sets, the limit procedure is expected to select only a discrete number of the $ V_0$'s and the corresponding pulses $ u_h(x; V_0)$. However, the ODE associated to the stationary problem for the PDE (3) is a dynamical system in $ {\bf R}^4$. A priori, there are no general techniques by which the existence of homoclinic orbits in such problems can be established. Furthermore, the stability problem yields a linear system that is neither of Sturm-Liouville type, nor self-adjoint, which implies that one should expect complex eigenvalues $ \lambda$. Moreover, the Sturm-Liouville equation (2) associated to the scalar limit indeed is a singular limit, the `perturbations' due to the coupling to the additional slow diffusion equation for $ V$ have a leading order effect on the eigenvalues. In other words: the coupling of $ U$ to a slow diffusion equation for $ V$ might be able to bring the unstable eigenvalue of the scalar limit problem to the stable $ \{ {\rm Re}(\lambda) \leqslant 0 \}$ half-plane.

As an example we present here two theorems that settle both the the existence problem and the stability problem in the special case of the classical Gierer-Mienhardt problem,

\begin{displaymath} \left\{ \begin{array}{rcccrcr} U_t & = & U_{xx} & + & [\fra... ...{xx} & + & \varepsilon ^2 [U^2 & - & \mu V] \end{array}\right. \end{displaymath} (4)

It should be remarked, however, that this case is less special than suggested by comparing (3) to (4). In fact, (4) can be seen as a prototypical system: under certain (generic) conditions equation (3) can be scaled into a normal form of the type (4) [3]. Essential in the derivation of the normal form is that both the $ U$-component $ U_h(x)$ as well as the $ V$-component $ V_h(x)$ of the pulse solution to (3) scale with a certain negative power of $ \varepsilon $ as $ \varepsilon \to 0$, i.e. the amplitude of the pulse $ (U_h(x), V_h(x))$ is in general asymptotically large [3].

Theorem 1. [3] For any $ \mu > 0$ there exists an $ \varepsilon _0 > 0$ such that for all $ 0 < \varepsilon < \varepsilon _0$ there exists a homoclinic pulse solution $ (U_h(x), V_h(x))$ to (4). Both components are of $ {\cal O}(\frac{1}{\varepsilon })$, i.e. both $ \lim_{\varepsilon \to 0} \varepsilon U_h(x)$ and $ \lim_{\varepsilon \to 0} \varepsilon V_h(x)$ exist (and are not 0). Moreover, $ \lim_{\varepsilon \to 0} \varepsilon U_h(x) = u_h(x)$, the homoclinic solution of the associated scalar limit problem (1) with $ F(U) = \frac{U^2}{V_0} - U$ and $ V_0 = \frac12 \sqrt{\mu}$.

See Figure 1. The proof of this theorem is based on the ideas of geometric singular perturbation theory, see [9].

Theorem 2. [3] The homoclinic pulse solution $ (U_h(x), V_h(x))$ of (4) is unstable for $ 0 < \mu < \mu_{\rm Hopf} = 0.36... + {\cal O}(\varepsilon )$, and spectrally stable for $ \mu > \mu_{\rm Hopf}$.

The proof of this result relies heavily on the Evans function approach as was developed in [1]. This method was extended in [4] to systems of the type (3), in the context of an explicit model problem - the Gray-Scott equation. Later, this method has been generalized, so that it is possible to consider the general equation (3). Note that one can go from a situation where the above intuitive argument (ii) is valid to a bifurcation that contradicts the intuition, by varying the parameter $ \mu$: if $ \mu > 0$ is small enough there is a unstable eigenvalue $ \lambda_1(\mu) \in {\bf R}$ that merges with the unstable eigenvalue $ \lambda_1 = \frac54$ of the scalar limit problem (2) in the limit $ \mu \to 0$ [3]. However, at a certain critical value $ \mu_{\rm complex} < \mu_{\rm Hopf}$ $ \lambda_1(\mu)$ merges with another real (positive) eigenvalue $ \lambda_2(\mu)$ and a pair of complex conjugated eigenvalues is formed. This pair crosses the imaginary axis as $ \mu$ passes through $ \mu_{\rm Hopf}$. Furthermore, it follows from the methods developed in [4,3] that the number of eigenvalues of the linearized stability problem associated to (4) is $ 4$, two of which can be complex, one more than that of the limit problem (2), which only has real eigenvalues. Hence, the relation between (4) and its scalar limit (1) with $ F(U) = \frac{U^2}{V_0} - U$ is more singular than expected at first sight.


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Next: Conclusion Up: Stabilization of pulses by Previous: Introduction